Author details

Definition 4. Following [20], p. 552, a bond portfolio BP is said to be a barbell strategy (barbell

Theorem 6. (see [19], Theorem 1). If Assumption 1 holds then the bond portfolio BP\* with the highest unanticipated rate of return is a barbell strategy built up of zero-coupon bearing bonds

, Bl with minimal and maximal dedicated durations. The weights xs and xl, expressing the

Comment 1. Suppose that instead of dedicated duration and dedicated convexity, we employ the classic notions of duration and convexity derived for additive shifts only. Then, vl � 1 and

Finally, another interesting question arises, to what extend does the dedicated duration of the best immunized portfolio BP\* differ from its Macaulay's counterpart? That is, what is the

It is easy to observe that when a shift a(t) affects the current interest rates in a similar manner at all or many points t1, t2, t3, …, tm, then there is a good chance that vs ≈ 1 ≈ vl, and consequently, the difference between the dedicated duration Dv and the classic one will be very small.

For a specific situation, when shifts a(t) of interest rates s(t) satisfied the "proportionality"

best immunizing bond portfolio was determined by means of Kuhn-Tucker conditions (pp. 139–140 in [21]). A formula for the resulting unanticipating rate of return was derived (pp.

Let us summarize what we have said so far. Each bond portfolio BP (a human body? or a human body organ?) generates cash at various dates t1, t2, t3, …, tm. What should (could) be substituted for cash (payments generated by a BP) in the medical setting remains an important open problem. Maybe, it is something related to a human body's performance; call this

In bond portfolio theory, the greater payouts generated by BP, the higher is the present value (PV) and future value (FV) of BP. An analogous statement is therefore expected in the medical context. Having settled what is Z, it would be probably easy to find out what is the counterpart

<sup>1</sup>þs tð Þ<sup>k</sup> <sup>¼</sup> constant (for details, see [21]), the maximal convexity and formula for the

, xl <sup>¼</sup> <sup>q</sup> � tsvs tlvl � tsvs

question arises of how much the weights given by Eq. (27) differ from the classic ones.

difference between Dv BP<sup>∗</sup> ð Þ¼ tswsvs <sup>þ</sup> tlwlvl and D BP<sup>∗</sup> ð Þ¼ tsws <sup>þ</sup> tlwl with vs <sup>¼</sup> a tð Þ<sup>s</sup>

. On the other hand, BP is said to be a focused strategy (focused

, are given by formulas:

, B<sup>2</sup> with significantly different dedicated

, xk ¼ 0 for k 6¼ s, k 6¼ l: (27)

tl�ts

, xl <sup>¼</sup> <sup>q</sup>�ts tl�ts

. The natural

a qð Þ, vl <sup>¼</sup> a tð Þ<sup>l</sup>

a qð Þ?

<sup>j</sup> are centered around

portfolio) if it is built up of two bonds, say B<sup>1</sup>

portfolio) if it consists of several bonds whose dedicated durations Dv

consequently Eq. (27) reduces to simpler, say classic, formulas xs <sup>¼</sup> tl�<sup>q</sup>

2

108 Immunization - Vaccine Adjuvant Delivery System and Strategies

amounts of payments resulting from B<sup>s</sup> and Bl

xs <sup>¼</sup> tlvl � <sup>q</sup> tlvl � tsvs

141–142) and illustrating with an example (p. 143).

<sup>1</sup> and Dv

duration of the liability (q in our context).

durations Dv

condition a tð Þ<sup>k</sup>

5. Concluding remarks

mysterious agent by Z.

Bs

Leszek Zaremba

Address all correspondence to: l.zaremba@vistula.edu.pl

Academy of Finance and Business Vistula, Warsaw, Poland
